An Efficient MMSE-Based Demodulator for MIMO Bit-Interleaved Coded Modulation

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1 in Proc IEEE GLOBECOM-04, Dallas (TX), Nov-Dec 004, pp Copyright IEEE 004 An Efficient MMSE-Based Demodulator for MIMO Bit-Interleaved Coded Modulation Domini Seethaler, Gerald Matz, and Franz Hlawatsch Institute of Communications and Radio-Frequency Engineering, Vienna University of Technology Gusshausstrasse 5/389, A-040 Vienna, Austria phone: , fax: , Currently on leave with Laboratoire des Signaux et Systèmes, École Supérieure d Électricité 3 Rue Joliot-Curie, F-990 Gif-sur-Yvette, France Abstract In bit-interleaved coded modulation (BICM) systems employing maximum-lielihood decoding, a demodulator (demapper) calculates a log-lielihood ratio (LLR) for each coded bit, which is then used as a bit metric for Viterbi decoding In the MIMO case, the computational complexity of LLR calculation tends to be excessively high, even if the logsum approximation is used Thus, there is a strong demand for efficient suboptimum MIMO-BICM demodulation algorithms with near-optimum performance Here, we propose an efficient MIMO-BICM demodulator that is derived by means of a Gaussian approximation for the post-detection interference Our derivation results in an MMSE equalizer followed by per-layer LLR calculation (ie, LLRs are calculated separately for each layer) The novel demodulator can be interpreted as an MMSE analogue of a recently proposed ZF-equalization based demodulator, as well as an extension of ZF-equalization based demodulation to correlated post-detection interference Because it performs per-layer LLR calculation, it has the same (low) computational complexity as the ZFequalization based demodulator Simulation results demonstrate that the performance of our demodulator is close to that of LLR calculation using all layers jointly, and significantly better than that of the ZF-equalization based demodulator I INTRODUCTION Bit-interleaved coded modulation (BICM) is a promising scheme for multiple-input multiple-output (MIMO) wireless communications Specifically, MIMO-BICM has been shown to outperform space-time trellis coding in fast-fading enviroments This is important because independent fast-fading MIMO channels provide a model for the individual subcarriers of MIMO systems using orthogonal frequency division multiplexing (OFDM) with frequency interleaving (eg,, 3), and for bloc-fading channels with temporal interleaving In BICM systems employing maximum-lielihood decoding, a demodulator (demapper) has to provide the channel decoder with bit metrics that are given by a log-lielihood ratio (LLR) for each coded bit In the MIMO case, LLR calculation is usually done by means of a complexity-reducing log-sum approximation ; this will be referred to as optimum log-sum approximation (OLSA) demodulation Because the LLRs are calculated jointly for all layers, OLSA demodulation tends to be excessively complex 3 Thus, there is a strong demand for efficient MIMO-BICM demodulation algorithms whose performance is close to that of OLSA demodulation Funding by FWF grant P556-N0 Such a demodulation algorithm with low complexity but near-optimum performance is proposed in this paper Our derivation of the novel demodulator is based on a Gaussian approximation for the post-detection interference, which is inspired and motivated by 4 We obtain a structure that consists of a minimum mean-square error (MMSE) equalizer with subsequent per-layer LLR calculation Thus, in contrast to OLSA demodulation, calculation of the LLRs is carried out for each layer separately The novel demodulator can be seen as an MMSE analogue of the zero-forcing (ZF) based demodulator recently proposed in 3, which uses ZF equalization followed by per-layer LLR calculation We will also show that the ZF-based demodulator, too, can be derived with a Gaussian approximation for the postdetection interference; however, in contrast to our approach, the interference has to be assumed uncorrelated Thus, our demodulator can also be seen as an extension of ZF-based demodulation to correlated post-detection interference, which explains its significant performance advantage over the ZFbased demodulator Because of the per-layer LLR calculation it employs, our demodulator has the same (low) computational complexity as the ZF-based demodulator This paper is organized as follows In the remainder of this section, we describe the system model and briefly review existing demodulation schemes The novel demodulator is derived and discussed in Section II Finally, Section III presents simulation results for fast-fading MIMO channels It is demonstrated that our demodulator achieves near-olsa performance for different alphabets and a wide range of SNRs, and that it significantly outperforms ZF-based demodulation A MIMO-BICM System Model We consider a flat-fading MIMO channel with M T transmit antennas and M R M T receive antennas (briefly termed an (M T,M R ) channel) We assume a spatial multiplexing system where for any given time instant n, theth data stream d n is directly transmitted on the th transmit antenna (or layer) This leads to the well-nown baseband model rn =Hn dn +wn, n =0,,N, () with the transmitted data symbol vector dn = ( d n d MT n ) T,theMR M T channel matrix Hn, the received Globecom /04/$ IEEE

2 dulator Demo MUX DEMUX Transmitter ~ bl Encoder Receiver rn Fig Π Π M T Π ΠM T Map Map Decoder MIMO system using bit-interleaved coded modulation dn vector rn = ( r n r MR n ) T, and the noise vector wn = ( w n w MR n ) T The data symbols d n are drawn from a complex symbol alphabet A and are assumed zero-mean with unit variance The noise components w n are assumed independent and circularly symmetric complex Gaussian with variance σw According to the MIMO-BICM system model of (see Fig ), a sequence of information bits bl is encoded using a convolutional code and cyclically demultiplexed into M T layers The coded bits of the th layer are interleaved (using an interleaver Π ) and subsequently mapped (using Gray labeling ) onto complex data symbols d n A At the receiver, the demodulator uses the received vector rn and nowledge of the channel Hn to calculate an LLR for each coded bit b (i) of the symbol vector dn The resulting sequences of LLRs are deinterleaved (using deinterleavers Π ) and multiplexed into a single stream, which is then used for soft-in Viterbi decoding (eg, 5) In the following, we concentrate on the demodulator that calculates the LLRs for a given symbol time index n For simplicity of notation, we will omit the time index and, thus, write () as r = Hd + w Letb (i) with i =,,log A denote the coded bits of the th layer, to which the symbol d A is associated via Gray labeling We assume that the code bits b (i) are equally liely and statistically independent B Review of MIMO-BICM Demodulation Algorithms OLSA Demodulator The LLR of b (i) is given by ( ) f(r b (i) = log =) f(r b (i) =0) = log d:d A i e σ w d:d A e i σ w 0 r Hd r Hd ^bl, () where A i b A denotes the set of all symbols a A whose label has b {0, } in bit position i With OLSA demodulation, the log-sum approximation is used to calculate the following approximate LLRs (eg, ): σw min r Hd min r Hd, (3) d:d A i 0 d:d A i which form the input to a soft-in Viterbi decoder OLSA demodulation achieves near-optimum performance However, its computational complexity is exponential in M T because the LLRs are calculated jointly for all layers, requiring the computation of A MT distances ZF-Equalization Based Demodulator A recently proposed alternative demodulation scheme 3 uses ZF equalization with subsequent per-layer LLR calculation (ie, LLRs are calculated separately for each layer) The ZF-equalized received vector is given by y ZF = H # r, (4) where H # =(H H H) H H is the pseudo-inverse 6 of H One obtains y ZF = d + w, (5) ie, the transmitted data vector d plus a transformed noise vector w = H # w whose covariance matrix is R w = σ w (H H H) (6) Motivated by (5) and (6), the ZF-based demodulator was proposed in 3 in an ad-hoc manner as,zf = σ w, min y ZF, a min y ZF, a, (7) where σ w, =(R w ), denotes the noise variance after ZF equalization for the th layer and y ZF, =(y ZF ) Because each layer is treated separately, only M T A scalar distances have to be calculated Thus, the complexity of (7) is only O(MT 3) (due to the calculation of y ZF), which is much smaller than the complexity of the OLSA demodulator in (3) However, the performance is significantly poorer (see Section III) II MMSE-BASED BICM DEMODULATION We will now derive and discuss the proposed demodulation technique In our derivation, we will use a Gaussian approximation for the post-detection interference to calculate approximate LLR values This is motivated and inspired by 4, where an iteratively updated Gaussian approximation for the post-detection interference was used in the context of multiuser detection Furthermore, we previously used a Gaussian interference approximation for developing a dynamic nulling-and-cancelling MIMO detector with improved performance 7 A The Gaussian Approximation We start by reformulating the LLR in terms of y ZF instead of r The exponent in () can be written as (eg, 8) The following derivation could also be performed without going into the ZF domain; however, in that case the calculations would be more involved Globecom /04/$ IEEE

3 σ w r Hd =(y ZF d) H R w (y ZF d) + r Hy ZF, and we can thus express the LLR () as d:d = log A i Inserting d:d A i 0 ( = log =) =0) e (yzf d)h R w e (yzf d)h R w ) a A i b (yzf d) (yzf d) =b) = A i b f(y ZF d =a) and using the log-sum approximation, we obtain further max log f(y ZF d =a) max log f(y ZF d =a) (8) Here, f(y ZF d =a) can be interpreted as the conditional probability density function (pdf) of the post-detection interference for the data symbol of interest, d Because of the Gaussianity of the noise, f(y ZF d = a) is a multivariate multimodal Gaussian mixture pdf To obtain a computationally efficient approximation to (8), we now approximate f(y ZF d = a) by a Gaussian pdf f(yzf d = a) with the same mean µ = E{y ZF d =a} and the same covariance C = cov{y ZF d =a}: f(y ZF d =a) = π MT det(c ) e (yzf µ ) H C (yzf µ ) (9) To find expressions of µ and C, we reformulate y ZF = d + w in (5) as M T y ZF = d e + d j e j + w, j= j where e is the th M T -dimensional unit vector We then obtain (recall that the d are independent with zero mean and var{d } =) µ = ae, C = I e e T + R w (0) The Gaussian pdf f(yzf d = a) is now completely determined, and the LLR in (8) is approximated according to Λ(i),MMSE = max log f(y ZF d =a) max log f(y ZF d =a) () The subscript MMSE in,mmse will be justified presently B Calculation of,mmse We will next derive a simple expression for,mmse that can be calculated very efficiently Inserting (9) and (0) into (), we obtain,mmse = min Q (a) min Q (a), () with the quadratic form Q (a) =(y ZF ae ) H C (y ZF ae ) = yzfc H y ZF Re { yzf H C e a } + a e T C e (3) The first term of the expression (3) does not depend on a and thus can be disregarded in () It remains to develop the second and third terms Applying the matrix inversion lemma 6 to C = ( I e e T + R w) and using (6), we obtain C = W I + e e T W W, with, (4) W =(I + R w ) = I + σ w(h H H) (this is termed Wiener estimator in 9) and with W, denoting the th diagonal element of W The Wiener estimator converts ZF equalization (4) into MMSE equalization 9, 0: y MMSE =(H H H + σwi) H H r = Wy ZF Using this result and (4), we obtain for the second and third terms in (3) yzf H C e = yzf H We + et We = y MMSE,, W, W, e T C e = e T We + et We W, = W, W, Furthermore, since the eigenvalues of W satisfy 0 λ W, <, the diagonal elements of W must satisfy 0 W, <, W, < It is then easily verified that () simplifies to,mmse = W, min ψ W (a) min ψ(a), (5), with the unbiased distance ψ (a) = y MMSE, W, a (6) Here, compared to the conventional distance y MMSE, a, the bias after MMSE equalization (defined as E{y MMSE, d d } ) is compensated through division of y MMSE, by W,, ie, E { y MMSE, } W, d = d It is important to note that, similarly to the ZF-based demodulator in (7) but in contrast to the OLSA demodulator in (3), the calculation of,mmse from y ZF is performed entirely within the th layer It can furthermore be shown that W, W, = SNR MMSE, Globecom /04/$ IEEE

4 Here, SNR MMSE, is the MMSE post-detection SNR of the th layer given by SNR MMSE, = MSE MMSE, (eg, ), where MSE MMSE, = E { y MMSE, d } is the minimum MSE of the th layer (eg, 3) Thus, (5) can be rewritten as,mmse = SNR MMSE, C Constant-Modulus Alphabets min ψ(a) min ψ(a) (7) For constant-modulus alphabets, ie, a =for all a A, (7) simplifies to,mmse = min y MMSE, a MSE MMSE, min y MMSE, a, (8) which is now formulated in terms of the biased distance y MMSE, a Even more simple expressions for,mmse are obtained for BPSK and 4-QAM (or QPSK) modulation: BPSK with Gray labeling: Λ (),MMSE = 4 MSE MMSE, Re{y MMSE, } ; (9) 4-QAM with Gray labeling:,mmse = MSE MMSE, Re{y MMSE, }, i =, MSE MMSE, Im{y MMSE, }, i = A result formally similar to (9) was obtained in a multiuser context in 4, however using a different approach D Discussion The proposed MIMO-BICM demodulator was derived by using a Gaussian approximation for the post-detection interference in the calculation of the LLRs Our final expression (7) (or (8) for constant-modulus alphabets), together with the expression of ψ (a) in terms of y MMSE, (see (6)), shows that the proposed demodulator consists of MMSE equalization and subsequent per-layer LLR calculation Because after MMSE equalization each layer is processed separately, the computational complexity of our demodulator is not higher than that of the ZF-based demodulator in (7) The structure of our demodulator is similar to both the OLSA demodulator in (3) and the ZF-based demodulator in (7) All three demodulators compute a difference of two distances, where one distance corresponds to the respective bit being 0 and the other corresponds to that bit being However, these distances are differently defined for the three demodulators Furthermore, the pre-factors in the approximate LLR expressions are different, too: With OLSA demodulation (3), the pre-factor is the reciprocal noise variance, /σ w With ZF-based demodulation (7), the pre-factor is the reciprocal post-equalization noise variance, /σ w, We note that σ w, can be shown to equal the ZF postdetection MSE, ie σ w, = MSE ZF, = E { y ZF, d } Finally, with our demodulator (7), the pre-factor is the MMSE post-detection SNR, SNR MMSE, ; in the special case of constant-modulus alphabets, the pre-factor in the alternative expression (8) is the reciprocal minimum post-detection MSE, /MSE MMSE, All pre-factors decrease with increasing noise power σw For constant-modulus alphabets, our demodulator (8) can be viewed as an MMSE analogue of the ZF-based demodulator (7), in which y ZF, is replaced by y MMSE, and σ w, =MSE ZF, is replaced by MSE MMSE, but not by the noise variance after MMSE equalization (note that in contrast to ZF equalization where σ w, = MSE ZF,, the noise variance after MMSE equalization is different from MSE MMSE, ) In fact, using the noise variance after MMSE equalization instead of MSE MMSE, in the pre-factor of (8) as motivated by the analogy to (7) would result in a significant performance degradation For general alphabets, our demodulator calculates the perlayer distances ψ (a) in (7) using the unbiased MMSE equalized components y MMSE, /W, instead of y MMSE, (cf (6)) For alphabets that are not constant-modulus, use of (8) instead of (7) would cause a slight performance degradation If we develop our demodulator by using the Gaussian approximation for the post-detection interference in (8), however with the correlations in y ZF neglected, the ZF-based demodulator (7) is obtained Indeed, it can be verified that with all nondiagonal elements of C = I e e T + R w set equal to zero, () becomes equivalent to (7) In this sense, then, our demodulator is an extension of the ZF-based demodulator to correlated post-detection interference This provides an explanation of the significant performance advantage of our demodulator over the ZF-based demodulator (as demonstrated in Section III) Fortunately, this performance advantage is obtained with no increase in complexity III SIMULATION RESULTS We now assess the performance of our demodulator by means of simulation results We considered a MIMO-BICM system using a rate-/ 64-state convolutional code with octal generators (33, 7) and random interleaving The MIMO channel had iid Gaussian matrix entries with unit variance To simulate fast fading, the channel was independently generated for each time instant We considered the proposed demodulator as well as the ZF-based and OLSA demodulators for comparison The Viterbi decoder used a tracebac depth of 35 and employed 6 bits for trellis termination Fig shows the bit-error rate (BER) of this MIMO-BICM system using the various demodulators versus the SNR for a (, ) channel and a (3, 3) channel Each figure part shows three sets of curves corresponding to BPSK, 4-QAM (QPSK), and 6-QAM modulation using Gray labeling The SNR is defined as E { Hd } /E { w } = M T /σ w Globecom /04/$ IEEE

5 BER BER (a) (b) (; ) (3; 3) BPSK 4-QAM 6-QAM OLSA proposed SNR db SNR db Fig BER performance of the proposed MMSE-based demodulator, the ZFbased demodulator, and the OLSA demodulator for BPSK, 4-QAM (QPSK), and 6-QAM modulation: (a) (,) channel, (b) (3,3) channel The following conclusions can be drawn from these results The performance of the proposed demodulator is very close to that of OLSA demodulation For all considered symbol alphabets, at a target BER of 0 4, our demodulator is within roughly db and db of OLSA performance for the (, ) channel and the (3, 3) channel, respectively Our demodulator significantly outperforms ZF-based demodulation even though its computational complexity is the same The performance gain over ZF-based demodulation is up to 4 db at a target BER of 0 4 In particular, for the (3, 3) channel, our demodulator for 4- QAM modulation achieves roughly the same performance as the ZF-based demodulator for BPSK modulation even though the data rate (in bits per channel use) is doubled The performance advantage of our demodulator over the ZF-based demodulator is strongest for small symbol alphabets Furthermore, a comparison of the results obtained for the (, ) channel and the (3, 3) channel suggests that for an increasing number of antennas, both the ZF performance advantage over the ZF-based demodulator and the performance loss relative to the OLSA demodulator grow larger On the other hand, the complexity savings achieved with our demodulator relative to the OLSA demodulator are larger for an increasing number of antennas and an increasing alphabet size IV CONCLUSIONS We have presented a novel, efficient demodulator for MIMO bit-interleaved coded modulation (BICM) systems Our development was based on a Gaussian approximation for the post-detection interference and resulted in a demodulation technique consisting of MMSE equalization and subsequent per-layer log-lielihood ratio (LLR) calculation The computational complexity of the novel MIMO-BICM demodulator is very low due to the per-layer processing it employs We showed that the proposed demodulator can be interpreted as an extension of a recently proposed ZF-equalization based demodulator to correlated post-detection interference This extension yields a substantial performance improvement without an increase in computational complexity We verified through simulations that the performance of our demodulator is close to that of LLR calculation using all layers jointly, and significantly better than that of the ZF-based demodulator REFERENCES G Caire, G Taricco, and E Biglieri, Bit-interleaved coded modulation, IEEE Trans Inf Theory, vol 44, pp , May 998 S H Müller-Weinfurtner, Coding approaches for multiple antenna transmission in fast fading and OFDM, IEEE Trans Signal Processing, vol 50, pp , Oct 00 3 M Butler and I Collings, A zero-forcing approximate log-lielihood receiver for MIMO bit-interleaved coded modulation, IEEE Commun Letters, vol 8, no, pp 05 07, Feb J Luo, K Pattipati, P Willett, and F Hasegawa, A PDA approach to CDMA multiuser detection, in Proc IEEE GLOBECOM-00, vol, San Antonio, TX, Nov 00, pp S Wilson, Digital Modulation and Coding Englewood Cliffs (NJ): Prentice Hall, G H Golub and C F Van Loan, Matrix Computations, 3rd ed Baltimore: Johns Hopins University Press, D Seethaler, H Artés, and F Hlawatsch, Dynamic nulling and cancelling with near-ml performance, in Proc IEEE ICASSP 004, vol IV, Montreal, Canada, May 004, pp B M Hochwald and S ten Brin, Achieving near-capacity on a multiple-antenna channel, IEEE Trans Inf Theory, vol 5, no 3, pp , March A Klein, G Kaleh, and P W Baier, Zero forcing and minimum mean square error equalization for multiuser detection in code-division multiple-access channels, IEEE Trans Veh Technol, vol 45, no, pp 76 87, May S M Kay, Fundamentals of Statistical Signal Processing: Estimation Theory Englewood Cliffs (NJ): Prentice Hall, 993 J M Cioffi, G P Dudevoir, M V Eyuboglu, and G D Forney, MMSE decision-feedbac equalizers and coding Part I: Equalization results, IEEE Trans Commun, vol 43, no 0, pp , Oct 995 R W Heath, S Sandhu, and A J Paulraj, Antenna selection for spatial multiplexing systems with linear receivers, IEEE Commun Letters, vol 5, no 4, pp 4 44, April 00 3 B Hassibi, A fast square-root implementation for BLAST, in Proc 34th Asilomar Conf Signals, Systems, Computers, Pacific Grove, CA, Nov/Dec 000, pp X Wang and H V Poor, Iterative (turbo) soft interference cancellation and decoding for coded CDMA, IEEE Trans Comm, vol 47, pp , July 999 Globecom /04/$ IEEE